FoldUnfold Table of Contents Arc Length of Curves in Three-Dimensional Space Examples 2 Example 1 Example 2 Arc Length of Curves in Three-Dimensional Space Examples 2 Recall from the Arc Length of Curves in Three-Dimensional Space that the arc length of a curve given by the vector equation $\vec{r}(t) = (x(t), y(t), z(t))$ that traces […]

## Arc Length of Curves in Three-Dimensional Space Examples 1

FoldUnfold Table of Contents Arc Length of Curves in Three-Dimensional Space Examples 1 Example 1 Example 2 Example 3 Arc Length of Curves in Three-Dimensional Space Examples 1 Recall from the Arc Length of Curves in Three-Dimensional Space that the arc length of a curve given by the vector equation $\vec{r}(t) = (x(t), y(t), z(t))$ […]

## Arc Length of Curves in Three-Dimensional Space

FoldUnfold Table of Contents Arc Length of Curves in Three-Dimensional Space Arc Length of Curves in Three-Dimensional Space We will now look at computing the arc length specified by a vector-valued function $\vec{r}(t)$ on an interval for which $t$ is in $I = [a, b]$. Let $C$ be a continuous and bounded curve specified by […]

## Parameterization of Curves in Three-Dimensional Space Examples 1

FoldUnfold Table of Contents Parameterization of Curves in Three-Dimensional Space Examples 1 Example 1 Example 2 Example 3 Parameterization of Curves in Three-Dimensional Space Examples 1 We saw some examples of parameterizing described curves in $\mathbb{R}^3$ on the Parameterization of Curves in Three-Dimensional Space page. We will now look at some more examples of parameterizing […]

## Parameterization of Curves in Three-Dimensional Space

FoldUnfold Table of Contents Parameterization of Curves in Three-Dimensional Space Example 1 Example 2 Example 3 Example 4 Parameterization of Curves in Three-Dimensional Space Sometimes we can describe a curve as an equation or as the intersections of surfaces in $\mathbb{R}^3$, however, we might rather prefer that the curve is parameterized so that we can […]

## Vector-Valued Functions – Velocity, Acceleration, and Speed

FoldUnfold Table of Contents Vector-Valued Functions – Velocity, Acceleration, and Speed Example 1 Vector-Valued Functions – Velocity, Acceleration, and Speed Sometimes we can use a vector-valued function $\vec{r}(t) = (x(t), y(t), z(t))$ to represent an object that traverses the curve $C$ traced by $\vec{r}(t)$ at time $t$. We can then define the velocity, acceleration, and […]

## Integrals of Vector-Valued Functions

FoldUnfold Table of Contents Integrals of Vector-Valued Functions Indefinite Integration of Vector-Valued Functions Definite Integration of Vector-Valued Functions Integrals of Vector-Valued Functions We have just looked at computing Derivatives of Vector-Valued Functions. We will now look at the vector-valued function analogue of integration. Let’s first start off with indefinite integration. Indefinite Integration of Vector-Valued Functions […]

## Derivative Rules for Vector Valued Functions Examples 1

FoldUnfold Table of Contents Derivative Rules for Vector Valued Functions Examples 1 Example 1 Example 2 Derivative Rules for Vector Valued Functions Examples 1 Recall from the Derivative Rules for Vector-Valued Functions page that if $\vec{u}(t) = (x_1(t), y_1(t), z_1(t))$ and $\vec{v}(t) = (x_2(t), y_2(t), z_2(t))$ are vector-valued functions that are differentiable for $t$ in […]

## Derivative Rules for Vector-Valued Functions

FoldUnfold Table of Contents Derivative Rules for Vector-Valued Functions Derivative Rules for Vector-Valued Functions We will now look at a bunch of rules for differentiating vector-valued function, all of which are analogous to that of differentiating real-valued functions. We will not prove all parts of the following theorem, but the reader is encouraged to attempt […]

## Derivatives of Vector-Valued Functions

FoldUnfold Table of Contents Derivatives of Vector-Valued Functions Example 1 Derivatives of Vector-Valued Functions We are now going to extend out concept of a derivative to vector-valued functions. Let $\vec{r}(t) = (x(t), y(t), z(t))$ be a vector-valued function defined for $t$ in the interval $I$ that traces out the curve $C$. Let $P$ and $Q$ […]

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